A Multidimensional Superposition Principle: Classical Solitons IV

This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition Formulae for nonlinear wave interactions. In the present research the technique of multiexpansions with constraints is considered for finding the above SFs and investigation of the related solitons. (The simplest case of such expansions technically is analogous to the so-called invariant truncated singular expansions.) As the applications, the soliton SFs of the MKdV_±, Kaup-Kupershmidt and new A_± equation are obtained for the bell-shape exponential solitons of the various families, algebraic solitons, and the configuration of the two noninteracting kinks. The linearized, parameterized versions of these SFs are investigated then, and the related analysis of the interactions is presented. The obtained results allow one to consider the one soliton solutions mentioned as the strong bound states of the simpler solitons. Concerning the results for the above concrete nonlinear PDEs, the approach being developed made it possible both to obtain the new results and to reveal new moments for the already known ones.     

Dynamical oscillations in nonlinear optical media

The spatial dynamics of pulses in Kerr media with parabolic index profile are examined. It is found that when diffraction and graded-index have opposite signs propagating pulses exhibit an oscillatory pattern, similar to a breathing behavior. Furthermore, if the pulse and the index profile are not aligned the pulse oscillates around the index origin with frequency that depends on the values of the diffraction and index of refraction. These oscillations are not observed when diffraction and graded-index share the same sign.     

Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes

A concept of time-reversed entropy per unit time is introduced in analogy with the entropy per unit time by Shannon, Kolmogorov, and Sinai. This time-reversed entropy per unit time characterizes the dynamical randomness of a stochastic process backward in time, while the standard entropy per unit time characterizes the dynamical randomness forward in time. The difference between the time-reversed and standard entropies per unit time is shown to give the entropy production of Markovian processes in nonequilibrium steady states.     

任意两个相干态的叠加态的相位分布和Wigner函数(英文)

从相位分布和Wigner函数两个方面研究了任意两个相干态|β〉and |mβeiδ〉的叠加态的量子统计性质.结果表明这种叠加态的非经典特性与β2,振幅系数m,相干态间的位相差δ以及叠加系数间的位相差都有关.当参量选择合适,这种叠加态存在着量子效应.计算了两个相干态等几率混合系综的相位分布和Wigner函数,经过与前者比较,结果表明由于相干项的存在,使得叠加态具有很好的量子力学行为.     

Algebraic Entropy of Birational Maps with Invariant Curves

The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this Letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.     

Modeling Thermostating, Entropy Currents, and Cross Effects by Dynamical Systems

A generalized multibaker map with periodic boundary conditions is shown to model boundary-driven transport, when the driving is applied by a perturbation of the dynamics localized in a macroscopically small region. In this case there are sustained density gradients in the steady state. A non-uniform stationary temperature profile can be maintained by incorporating a heat source into the dynamics, which deviates from the one of a bulk system only in a (macroscopically small) localized region such that a heat (or entropy) flux can enter an attached thermostat only in that region. For these settings the relation between the average phase-space contraction, the entropy flux to the thermostat and irreversible entropy production is clarified for stationary and non-stationary states. In addition, thermoelectric cross effects are described by a multibaker chain consisting of two parts with different transport properties, modeling a junction between two metals.     

Coherent Transport and Dynamical Entropy for Fermionic Systems

This paper consists of two parts. First we set up a general scheme of local traps in a homogeneous deterministic quantum system. The current of particles caught by the trap is linked to the dynamical behaviour of the trap states. In this way, transport properties in a homogeneous system are related to spectral properties of a coherent dynamics. Next we apply the scheme to a system of Fermions in the one-particle approximation. We obtain in particular lower bounds for the dynamical entropy in terms of the current induced by the trap.     

任意两个相干态的叠加态的相位分布和Wigner函数

从相位分布和Wigner函数两个方面研究了任意两个相干态|β〉 and |mβeiδ〉的叠加态的量子统计性质.结果表明这种叠加态的非经典特性与β2,振幅系数m,相干态间的位相差δ以及叠加系数间的位相差都有关.当参量选择合适,这种叠加态存在着量子效应.计算了两个相干态等几率混合系综的相位分布和Wigner函数,经过与前者比较,结果表明由于相干项的存在,使得叠加态具有很好的量子力学行为.     

Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps: I

Departing from the symbolic dynamics, we study natural group action on spaces of holomorphic maps and complex subvarieties.     

Dynamical Entropy of Generalized Quantum Markov Chains on Gauge-invariant C *-algebras

We prove that the mean entropy and the dynamical entropy are equal for generalized quantum Markov chains on gauge-invariant C ^*-algebras.     

A Finite Partition in a Dynamical System with Entropy Depending on the Size

A finite partition X in a C*-dynamical system has the Alicki–Fannes entropy depending on the size of X so that the dynamical entropy is . Examples are given as Cuntz's canonical endomorphism, the inner automorphism Adu on the crossed product of a quantum spin chain by the shift, and the free shift.     

动态关联和110Cd的集体态

提出了动态关联的概念,并用这种关联讨论了¹¹⁰Cd的集体态. 同时还考虑了该核素中的g玻色子自由度,并在sdf玻色子的框架下讨论了它的八极态.     

Superposition, Entropy and Schmidt Decomposition of States

Superposition and entropy are compared using the language of the logic of quantum mechanics. It is pointed out that a finite value of the relative quantum entropy of states implies a superposition relation between the states themselves. The superposition relation is then studied by comparing the pure state of the compound system with the product of the reduced states and an intermediate Schmidt state. All the corresponding relative quantum entropies are evaluated in terms of the Schmidt coefficients of the global pure state. Some of the results are extended in case the compound system is in a state represented by a general density operator.     

Quantifying Information without Entropy: Identifying Intermittent Disturbances in Dynamical Systems

A system’s response to disturbances in an internal or external driving signal can be characterized as performing an implicit computation, where the dynamics of the system are a manifestation of its new state holding some memory about those disturbances. Identifying small disturbances in the response signal requires detailed information about the dynamics of the inputs, which can be challenging. This paper presents a new method called the Information Impulse Function (IIF) for detecting and time-localizing small disturbances in system response data. The novelty of IIF is its ability to measure relative information content without using Boltzmann’s equation by modeling signal transmission as a series of dissipative steps. Since a detailed expression of the informational structure in the signal is achieved with IIF, it is ideal for detecting disturbances in the response signal, i.e., the system dynamics. Those findings are based on numerical studies of the topological structure of the dynamics of a nonlinear system due to perturbated driving signals. The IIF is compared to both the Permutation entropy and Shannon entropy to demonstrate its entropy-like relationship with system state and its degree of sensitivity to perturbations in a driving signal.     

Entropy and ionic conductivity

It is known that the ionic conductivity can be obtained by using the diffusion constant and the Einstein relation. We derive it here by extracting it from the steady electric current which we calculate in three ways, using statistics analysis, an entropy method, and an entropy production approach.     

Entropy changes for steady-state fluctuations

The dissipative steady state far from equilibrium and subject to a slow modulation of external parameters is analyzed. It is shown that the time-integrated energy dissipation consists of three terms. The first of these is irreversible and consists of the time-integrated dissipation of the sequence of exact steady states defined by the externally controlled parameters traversed during the modulation. The second term is reversible and reflects the fact that the dissipation of the time-dependent modulated system, as calculated in a macroscopic way from ensemble averages, is not the same as the dissipation of a sequence of exact steady states. The third term is also reversible and relates to the ensemble dispersion in changes in stored energy during the modulation. If the system has a single degree of freedom and narrow fluctuations, then these fluctuations can be characterized by an effective temperature T_N. The third term can then be shown to be equal toT _N dS, whereS is the entropy calculated from the distribution function by the usual definition.     

Dynamical quantities and their numerical analysis by saddle periodic orbits

We study estimators for dynamical quantities such as the topological entropy and the topological pressure which are based on numerical computations on periodic orbits. For the particular case of the Hénon family for some parameter ranges we find a reasonable convergence of the entropy, the pressure, and Birkhoff averages of test functions. However, pointing out possible pitfalls of such an analysis, we show that the evaluation by means of saddle orbits alone can be misleading if, for example, chaotic saddles and attractors coexist.     

Computing the Topological Entropy for Piecewise Monotonic Maps on the Interval

A new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented. It uses a transition matrix associated with the transformation. For this matrix we give a spectral theorem. This can be used for an estimation of the accuracy of the algorithm.     

Entropy of the Binomial Field Interacting with the Two Entangled Atoms

In this paper, we investigate entropy properties of the single-mode binomial field interacting with the two entangled two-level atoms. It is found that the different initial conditions lead to the different evolutions of entropy of the binomial field.